Subgroups — C₂×#13092981 #15686703

Order 2944.

1896 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₉₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092981	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10625111	C₂	1	2	yes	maximal, complemented
4	C₂×#10625118	C₂	1	2	yes	maximal, complemented
5	C₂×#10625101	C₂	1	4	yes	maximal, complemented
6	C₂×#10625108	C₂	1	4	yes	maximal, complemented
7	C₂×#10625097	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₂₃	C₂²	1	2	yes	—
10	C₂×#41×C₂₃	—	2	8	no	—
11	C₂×#41×C₂₃	C₂²	1	8	yes	complemented
12	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
14	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
15	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₂₃	C₂²	1	2	yes	complemented
17	C₂²×C₂²⋊D₂₃	C₂²	1	8	yes	complemented
18	C₂²×D₉₂	—	2	8	no	—
19	C₂²×D₉₂	C₂²	1	2	yes	—
20	C₂²×D₉₂	C₂²	1	2	yes	—
21	C₂²×D₉₂	C₂²	1	8	yes	complemented
22	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
23	#10625118	C₂²	1	16	yes	complemented
24	#10625111	C₂²	1	16	yes	complemented
25	#10625108	C₂²	1	32	yes	complemented
26	#10625101	C₂²	1	32	yes	complemented
27	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	8	yes	complemented
28	#10625097	C₂²	1	8	yes	complemented
29	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	complemented
30	C₂×C₄×Dic₂₃	C₂²	1	2	yes	complemented
31	C₂⁴×C₂₃	C₂³	1	2	yes	—
32	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
33	C₂³×D₂₃	—	2	8	no	—
34	C₂³×D₂₃	—	4	8	no	—
35	C₂³×D₂₃	C₂³	1	4	yes	—
36	C₂³×D₂₃	C₂³	1	8	yes	—
37	#41×C₂₃	—	2	32	no	—
38	C₂×Q₈×C₂₃	—	2	4	no	—
39	C₂×Q₈×C₂₃	C₂³	1	4	yes	complemented
40	C₂×D₄×C₂₃	—	2	8	no	—
41	C₂×D₄×C₂₃	—	2	4	no	—
42	C₂×D₄×C₂₃	C₂³	1	8	yes	—
43	C₂×D₄×C₂₃	C₂³	1	4	yes	complemented
44	C₂²×C₄×C₂₃	—	2	8	no	—
45	C₂²×C₄×C₂₃	D₄	1	8	yes	complemented
46	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
50	C₂²×C₄×C₂₃	C₂³	1	8	yes	complemented
51	C₂×C₂²⋊D₂₃	—	4	16	no	—
52	C₂×C₂²⋊D₂₃	C₂³	1	32	yes	complemented
53	C₂²×Dic₂₃	C₂³	1	4	yes	—
54	C₂×D₉₂	—	2	8	no	—
55	C₂×D₉₂	—	2	8	no	—
56	C₂×D₉₂	—	2	32	no	—
57	C₂×D₉₂	C₂³	1	8	yes	—
58	C₂×D₉₂	C₂³	1	8	yes	—
59	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
60	(C₂×D₂₃)⋊C₄	C₂³	1	32	yes	complemented
61	C₄⋊Dic₂₃	C₂³	1	8	yes	complemented
62	C₄×Dic₂₃	C₂³	1	8	yes	complemented
63	C₂³×C₂₃	—	2	4	no	—
64	C₂³×C₂₃	—	2	4	no	—
65	C₂³×C₂₃	—	2	8	no	—
66	C₂³×C₂₃	C₂×D₄	1	4	yes	—
67	C₂³×C₂₃	C₂⁴	1	4	yes	—
68	C₂³×C₂₃	C₂⁴	1	2	yes	—
69	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
70	C₂²×D₂₃	—	2	16	no	—
71	C₂²×D₂₃	—	2	8	no	—
72	C₂²×D₂₃	—	2	16	no	—
73	C₂²×D₂₃	—	4	16	no	—
74	C₂²×D₂₃	—	4	32	no	—
75	C₂²×D₂₃	C₂⁴	1	8	yes	—
76	Q₈×C₂₃	—	2	8	no	—
77	D₄×C₂₃	—	2	16	no	—
78	D₄×C₂₃	—	2	8	no	—
79	C₂×C₄×C₂₃	—	2	4	no	—
80	C₂×C₄×C₂₃	—	2	16	no	—
81	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
82	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
83	C₂×C₄×C₂₃	C₂×D₄	1	16	yes	complemented
84	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
85	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
86	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
87	C₂²⋊D₂₃	—	4	32	no	—
88	C₂×Dic₂₃	—	2	8	no	—
89	C₂×Dic₂₃	C₂⁴	1	8	yes	—
90	D₉₂	—	2	16	no	—
91	D₉₂	—	2	16	no	—
92	C₂×#535	—	23	1	no	maximal, Sylow
93	C₂²×C₂₃	—	2	8	no	—
94	C₂²×C₂₃	—	2	4	no	—
95	C₂²×C₂₃	—	2	4	no	—
96	C₂²×C₂₃	—	2	8	no	—
97	C₂²×C₂₃	—	4	8	no	—
98	C₂²×C₂₃	C₂²×D₄	1	4	yes	—
99	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
100	C₂²×C₂₃	H(5,𝔽₂)	1	4	yes	—
101	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
102	C₂×D₂₃	—	2	8	no	—
103	C₂×D₂₃	—	2	16	no	—
104	C₂×D₂₃	—	4	32	no	—
105	C₂×D₂₃	—	4	16	no	—
106	C₄×C₂₃	—	2	4	no	—
107	C₄×C₂₃	C₂²×D₄	1	4	yes	—
108	C₄×C₂₃	C₂²×D₄	1	4	yes	—
109	Dic₂₃	—	2	8	no	—
110	C₂²×#41	D₂₃	1	1	yes	characteristic, complemented
111	C₂³×D₄	—	23	1	no	—
112	#535	—	23	16	no	—
113	C₂×#127	—	23	2	no	—
114	C₂×#124	—	23	2	no	—
115	C₂×#121	—	23	4	no	—
116	C₂×#120	—	23	4	no	—
117	C₂×#117	—	23	1	no	—
118	C₂×C₂₃	—	2	4	no	—
119	C₂×C₂₃	—	4	4	no	—
120	C₂×C₂₃	#535	1	4	yes	complemented
121	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
122	C₂×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
123	D₂₃	—	4	8	no	—
124	C₂⁵	—	23	2	no	—
125	C₂×#41	—	2	8	no	—
126	C₂×#41	C₂×D₂₃	1	8	yes	complemented
127	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
128	C₂²×D₄	—	23	8	no	—
129	C₂²×D₄	—	23	2	no	—
130	C₂²×D₄	—	23	2	no	—
131	C₂²×D₄	—	23	8	no	—
132	C₂²×D₄	—	46	8	no	—
133	C₂²×D₄	C₂×D₂₃	1	2	yes	—
134	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic, complemented
135	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
136	C₂³×C₄	C₂×D₂₃	1	2	yes	complemented
137	#127	—	23	16	no	—
138	#124	—	23	16	no	—
139	#121	—	23	32	no	—
140	#120	—	23	32	no	—
141	#117	—	23	8	no	—
142	C₂×C₄⋊C₄	—	23	2	no	—
143	C₂×C₂²⋊C₄	—	23	2	no	—
144	C₂×C₂²⋊C₄	—	23	8	no	—
145	C₂×C₄²	—	23	2	no	—
146	C₂₃	C₂×#535	1	1	yes	characteristic, complemented, Sylow
147	C₂⁴	—	23	4	no	—
148	C₂⁴	—	23	8	no	—
149	C₂⁴	—	46	8	no	—
150	C₂⁴	—	92	8	no	—
151	C₂⁴	C₂²×D₂₃	1	2	yes	—
152	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
153	#41	—	2	32	no	—
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	C₂²×D₂₃	1	4	yes	complemented
156	C₂×D₄	—	2	8	no	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	—	23	32	no	—
159	C₂×D₄	—	23	8	no	—
160	C₂×D₄	—	23	8	no	—
161	C₂×D₄	—	46	8	no	—
162	C₂×D₄	—	46	8	no	—
163	C₂×D₄	—	46	32	no	—
164	C₂×D₄	—	92	16	no	—
165	C₂×D₄	C₂²×D₂₃	1	8	yes	—
166	C₂×D₄	C₂²×D₂₃	1	4	yes	complemented
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	—	23	4	no	—
169	C₂²×C₄	C₂²⋊D₂₃	1	8	yes	complemented
170	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
171	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
174	C₂²×C₄	C₂²×D₂₃	1	8	yes	complemented
175	C₄⋊C₄	—	23	8	no	—
176	C₂²⋊C₄	—	23	8	no	—
177	C₂²⋊C₄	—	23	32	no	—
178	C₄²	—	23	8	no	—
179	C₂³	—	2	4	no	—
180	C₂³	—	2	8	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	23	8	no	—
183	C₂³	—	46	16	no	—
184	C₂³	—	46	16	no	—
185	C₂³	—	46	8	no	—
186	C₂³	—	92	16	no	—
187	C₂³	—	92	32	no	—
188	C₂³	C₂×C₂²⋊D₂₃	1	4	yes	—
189	C₂³	C₂³×D₂₃	1	4	yes	—
190	C₂³	C₂³×D₂₃	1	2	yes	—
191	C₂³	C₂³×D₂₃	1	1	yes	characteristic
192	Q₈	—	2	8	no	—
193	D₄	—	2	16	no	—
194	D₄	—	2	8	no	—
195	D₄	—	46	16	no	—
196	D₄	—	46	16	no	—
197	D₄	—	92	32	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	16	no	—
200	C₂×C₄	—	23	8	no	—
201	C₂×C₄	—	46	8	no	—
202	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
203	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
204	C₂×C₄	C₂×C₂²⋊D₂₃	1	16	yes	complemented
205	C₂×C₄	C₂³×D₂₃	1	4	yes	—
206	C₂×C₄	C₂³×D₂₃	1	2	yes	—
207	C₂×C₄	C₂³×D₂₃	1	2	yes	—
208	C₂²	—	2	8	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	8	no	—
212	C₂²	—	4	8	no	—
213	C₂²	—	46	8	no	—
214	C₂²	—	46	16	no	—
215	C₂²	—	92	32	no	—
216	C₂²	—	92	16	no	—
217	C₂²	#10625160	1	4	yes	—
218	C₂²	C₂²×C₂²⋊D₂₃	1	4	yes	—
219	C₂²	C₂²×C₂²⋊D₂₃	1	2	yes	—
220	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	46	8	no	—
223	C₄	C₂²×C₂²⋊D₂₃	1	4	yes	—
224	C₄	C₂²×C₂²⋊D₂₃	1	4	yes	—
225	C₂	—	2	4	no	—
226	C₂	—	4	4	no	—
227	C₂	—	92	8	no	—
228	C₂	#13092981	1	4	yes	direct product factor
229	C₂	C₂³×C₂²⋊D₂₃	1	1	yes	characteristic
230	C₂	C₂×#10625160	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂²×#41×C₂₃	C₂
socle	C₂³×C₂₃	C₂⁴
hypercenter	C₂²×#41	D₂₃
hypocenter	C₂₃	C₂×#535
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×D₉₂	C₂	semidirect
2. #13092981	C₂	direct (N × H)
3. C₂×#10625111	C₂	semidirect
4. C₂×#10625118	C₂	semidirect
5. C₂×#10625101	C₂	semidirect
6. C₂×#10625108	C₂	semidirect
7. C₂×#10625097	C₂	semidirect
8. C₂²×#41×C₂₃	C₂	semidirect
11. C₂×#41×C₂₃	C₂²	semidirect
12. C₂²×Q₈×C₂₃	C₂²	semidirect
14. C₂²×D₄×C₂₃	C₂²	semidirect
16. C₂³×C₄×C₂₃	C₂²	semidirect
17. C₂²×C₂²⋊D₂₃	C₂²	semidirect
21. C₂²×D₉₂	C₂²	semidirect
23. #10625118	C₂²	semidirect
24. #10625111	C₂²	semidirect
25. #10625108	C₂²	semidirect
26. #10625101	C₂²	semidirect
27. C₂×(C₂×D₂₃)⋊C₄	C₂²	semidirect
28. #10625097	C₂²	semidirect
29. C₂×C₄⋊Dic₂₃	C₂²	semidirect
30. C₂×C₄×Dic₂₃	C₂²	semidirect
39. C₂×Q₈×C₂₃	C₂³	semidirect
43. C₂×D₄×C₂₃	C₂³	semidirect
45. C₂²×C₄×C₂₃	D₄	semidirect
50. C₂²×C₄×C₂₃	C₂³	semidirect
52. C₂×C₂²⋊D₂₃	C₂³	semidirect
60. (C₂×D₂₃)⋊C₄	C₂³	semidirect
61. C₄⋊Dic₂₃	C₂³	semidirect
62. C₄×Dic₂₃	C₂³	semidirect
83. C₂×C₄×C₂₃	C₂×D₄	semidirect
110. C₂²×#41	D₂₃	semidirect
120. C₂×C₂₃	#535	semidirect
126. C₂×#41	C₂×D₂₃	semidirect
127. C₂²×Q₈	C₂×D₂₃	semidirect
134. C₂²×D₄	C₂×D₂₃	semidirect
136. C₂³×C₄	C₂×D₂₃	semidirect
146. C₂₃	C₂×#535	semidirect
155. C₂×Q₈	C₂²×D₂₃	semidirect
166. C₂×D₄	C₂²×D₂₃	semidirect
169. C₂²×C₄	C₂²⋊D₂₃	semidirect
174. C₂²×C₄	C₂²×D₂₃	semidirect
204. C₂×C₄	C₂×C₂²⋊D₂₃	semidirect
228. C₂	#13092981	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×D₉₂	2
#13092981	2
C₂×#10625111	2
C₂×#10625118	2
C₂×#10625101	2
C₂×#10625108	2
C₂×#10625097	2
C₂²×#41×C₂₃	2
C₂×#535	23

Subgroups — C2×#13092981 #15686703

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092981 #15686703